Multi-Task Learning and Matrix Regularization

Size: px

Start display at page:

Download "Multi-Task Learning and Matrix Regularization"

Luke Hart
6 years ago
Views:

1 Multi-Task Learning and Matrix Regularization Andreas Argyriou Department of Computer Science University College London

2 Collaborators T. Evgeniou (INSEAD) R. Hauser (University of Oxford) M. Herbster (University College London) A. Maurer (Stemmer Imaging) C.A. Micchelli (SUNY Albany) M. Pontil (University College London) Y. Ying (University of Bristol) 1

3 Main Themes Machine learning Convex optimization Sparse recovery 2

4 Outline Multi-task learning and related problems Matrix learning and an alternating algorithm Extensions of the method Multi-task representer theorems Kernel hyperparameter learning; convex kernel learning 3

5 Supervised Learning (Single-Task) m examples are given: (x 1, y 1 ),...,(x m, y m ) X Y Predict using a function f : X Y Want the function to generalize well over the whole of X Y Includes regression, classification etc. Task = probability measure on X Y 4

6 Multi-Task Learning Tasks t = 1,..., n m examples per task are given: (x t1, y t1 ),...,(x tm, y tm ) X Y Predict using functions f t : X Y, t = 1,..., n When the tasks are related, learning the tasks jointly should perform better than learning each task independently Especially important when few data points are available per task (small m); in such cases, independent learning is not successful 5

7 Multi-Task Learning (contd.) One goal is to learn what structure is common across the n tasks Want simple, interpretable models that can explain multiple tasks Want good generalization on the n given tasks but also on new tasks (transfer learning) Given a few examples from a new task t, {(x t 1, y t 1),...,(x t l, y t l)}, want to learn f t using just the learned task structure 6

8 Learning Theoretic View: Environment of Tasks Environment = probability distribution on a set of learning tasks [Baxter, 1996] To sample a task-specific sample from the environment draw a function f t from the environment generate a sample {(x t1, y t1 ),...,(x tm, y tm )} (X Y) m using f t Multi-task learning means learning a common hypothesis space 7

9 Learning Theoretic View (contd.) Baxter s results: As n (#tasks) increases, m (#examples per task needed) decreases as O( 1 n ) Once we have learned a hypothesis space H, we can use it to learn a new task drawn from the same environment; the sample complexity depends on the log-capacity of H Other results: Task relatedness due to input transformations: improved multi-task bounds in some cases [Ben-David & Schuller, 2003] Using common feature maps (bounded linear operators): error bounds depend on Hilbert-Schmidt norm [Maurer, 2006] 8

10 Multi-Task Applications Multi-task learning is ubiquitous Human intelligence relies on transfering learned knowledge from previous tasks to new tasks E.g. character recognition (very few examples should be needed to recognize new characters) Integration of medical / bioinformatics databases 9

11 Multi-Task Applications (contd.) Marketing databases, collaborative filtering, recommendation systems (e.g. Netflix); task = product preferences for each person 10

12 Multi-Task Applications (contd.) Multiple object classification in scenes: an image may contain multiple objects; learning common visual features enhances performance 11

13 Related Problems Sparse coding (some images share common basis images) Vector-valued / structured output Multi-class problems Regression with grouped variables, multifactor ANOVA in statistics; (selection of groups of variables) Multi-task learning is a broad problem; no single method can solve everything 12

14 Learning Multiple Tasks with a Common Kernel Let X IR d, Y IR and let us learn n linear functions f t (x) = w t, x t = 1,..., n (we ignore nonlinearities for the moment) Want to impose common structure / relatedness across tasks Idea: use a common linear kernel for all tasks K(x, x ) = x,d x (where D 0) 13

15 Learning Multiple Tasks with a Common Kernel For every t = 1,..., n solve min w t IR d m E ( w t, x ti, y ti ) + γ w t, D 1 w t i=1 Adding up, we obtain the equivalent problem min w 1,...,w n IR d n t=1 m E ( w t, x ti, y ti ) + γ i=1 n w t, D 1 w t t=1 14

16 Learning Multiple Tasks with a Common Kernel For multi-task learning, we want to learn the common kernel from a convex set of kernels: inf w 1,...,w n IR d D 0, tr(d) 1 n t=1 m E ( w t, x ti, y ti ) + γ tr(w D 1 W) (MT L) i=1 We denote W = w 1... w n n w t, D 1 w t t=1 15

17 Learning Multiple Tasks with a Common Kernel Jointly convex problem in (W, D) The constraint tr(d) 1 is important Fixing W, the optimal D(W) is D(W) (WW ) 2 1 (D(W) is usually not in the feasible set because of the inf) Once we have learned ˆD, we can transfer it to learning of a new task t min w IR d m E ( w, x t i, y t i) + γ w, ˆD 1 w i=1 16

18 Alternating Minimization Algorithm Alternating minimization over W (supervised learning) and D (unsupervised correlation of tasks). Initialization: set D = I d d d while convergence condition is not true do for t = 1,..., n learn w t independently by minimizing m E( w, x ti, y ti ) + γ w, D 1 w i=1 end for set D = (WW ) 1 2 end while tr(ww )

19 Alternating Minimization (contd.) Each w t step is a regularization problem (e.g. SVM, ridge regression etc.) It does not require computation of the (pseudo)inverse of D Each D step requires an SVD; this is usually the most costly step 18

20 Alternating Minimization (contd.) The algorithm (with some perturbation) converges to an optimal solution n m E( w t, x ti, y ti ) + γ tr ( D 1 (WW + εi) ) (R ε ) min w 1,...,w n IR d D 0, tr(d) 1 t=1 i=1 Theorem. An alternating algorithm for problem (R ε ) has the property that its iterates ( W (k), D (k)) converge to the minimizer of (R ε ) as k. Theorem. Consider a sequence ε l 0 + and let (W l, D l ) be the minimizer of (R εl ). Then any limiting point of the sequence {(W l, D l )} is an optimal solution of the problem (MT L). Note: the starting value of D does not matter 19

21 Alternating Minimization (contd.) η = 0.05 η = 0.03 η = 0.01 Alternating Alternating η = 0.05 objective function 26 seconds #iterations (green = alternating) #tasks (blue = alternating) Compare computational cost with a gradient descent approach (η := learning rate) 20

22 Alternating Minimization (contd.) Typically fewer than 50 iterations needed in experiments At least an order of magnitude fewer iterations than gradient descent (but cost per iteration is larger) Scales better with the number of tasks Both methods require SVD (costly if d is large) Alternative algorithms: SOCP methods [Srebro et al. 2005, Liu and Vandenberghe 2008], gradient descent on matrix factors [Rennie & Srebro 2005], singular value thresholding [Cai et al. 2008] 21

23 Trace Norm Regularization Eliminating D in optimization problem (MT L) yields min W IR d n n t=1 m E( w t, x ti, y ti ) + γ W 2 tr (T R) i=1 The trace norm (or nuclear norm) W tr is the sum of the singular values of W There has been recent interest in trace norm / rank problems in matrix factorization, statistics, matrix completion etc. [Cai et al. 2008, Fazel et al. 2001, Izenman 1975, Liu and Vandenberghe 2008, Srebro et al. 2005] 22

24 Trace Norm vs. Rank Problem (T R) is a convex relaxation of the problem min W IR d n n t=1 m E( w t, x ti, y ti ) + γ rank(w) i=1 NP-hard problem (at least as hard as Boolean LP) Rank and trace norm correspond to L 0, L 1 on the vector of singular values Multi-task intuition: we want the task parameter vectors w t to lie on a low dimensional subspace 23

25 Connection to Group Lasso Problem (MT L) is equivalent to n m E( a t, U x ti, y ti ) + γ A 2 2,1 min A IR d n U IR d d, U U=I where A 2,1 := d i=1 t=1 n i=1 a 2 it t=

26 Experiment (Computer Survey) Consumers ratings of products [Lenk et al. 1996] 180 persons (tasks) 8 PC models (training examples); 4 PC models (test examples) 13 binary input variables (RAM, CPU, price etc.) + bias term Integer output in {0,..., 10} (likelihood of purchase) The square loss was used 25

27 Experiment (Computer Survey) Test error Eig(D) #tasks Performance improves with more tasks (for learning of the tasks independently, error = 16.53) A single most important feature shared by all persons 26

28 Experiment (Computer Survey) 0.25 u TE RAM SC CPU HD CD CA CO AV WA SW GU PR Method RMSE Alternating Alg Hierarchical Bayes [Lenk et al.] 1.90 Independent 3.88 Aggregate 2.35 Group Lasso 2.01 The most important feature (eigenvector of D) weighs technical characteristics (RAM, CPU, CD-ROM) vs. price 27

29 Spectral Regularization Generalize (MT L): inf w 1,...,w n IR d D 0, tr(d) 1 n t=1 m E ( w t, x ti, y ti ) + γ tr(w F(D)W) i=1 where F is a spectral matrix function, f : (0, + ) (0, + ), or F(UΛU ) = U diag[f(λ 1 ),...,f(λ d )] U min W IR d n n t=1 m E ( w t, x ti, y ti ) + γ Ω(W) i=1 It can be shown that Ω(W) is a function of the singular values of W 28

30 Spectral Regularization (contd.) In particular, if f(λ) = λ 1 2 p, p (0,2], we have Ω(W) = W 2 p where W p is the Schatten L p (pre)norm of the singular values of W Theorem. The regularizer tr(w F(D)W) is jointly convex if and only if 1 f is matrix concave of order d, that is, ( ) ( ) ( ) µ (A) + (1 µ) (B) (µa + (1 µ)b) F F F for all A, B 0, µ [0,1] Spectral problems appear also in graph applications, control theory etc. 29

31 Learning Groups of Tasks Assume heterogeneous environment, i.e. K low dimensional subspaces Learn a partition of tasks in K groups inf D 1,...,D K 0 tr(d k ) 1 n t=1 K min k=1 min w t IR d { m i=1 E ( w t, x ti, y ti ) + γ w t, D 1 k w t } The representation learned is ( ˆD 1,..., ˆD K ); we can transfer this representation to easily learn a new task Non-convex problem; we use stochastic gradient descent 30

32 Experiment (Character Recognition - Projection on Image Halves) 6 vs. 1 task (on the right half) Binary classification tasks on images One half of the image contains the relevant character, the other half contains a randomly chosen character Two groups of tasks (with probabilities 50-50%): projection on left half - projection on right half 31

33 Experiment (Character Recognition - Projection on Image Halves) Training set contains pairs of alphabetic characters 1000 tasks, 10 examples per task Wish to obtain a representation that represents rotation invariance on either half of the image Wish to transfer this representation to pairs of digits 32

34 Experiment (Character Recognition - Projection on Image Halves) Independent K = 1 K = Transfer error for different methods D 1 D 2 All digits (left & right) 48.2% 51.8% Left 99.2% 0.8% Right 1.4% 98.6% Training data 50.7% 49.3% Assignment of tasks in groups (with K = 2) 33

35 Experiment (Character Recognition - Projection on Image Halves) D D 1 D 2 Dominant eigenvectors of D (for K = 1) and D 1, D 2 (for K = 2) 34

36 Experiment (Character Recognition - Projection on Image Halves) Spectrum of D learned with K = 1 (left) Spectra of D 1 (middle) and D 2 (right) learned with K = 2 35

37 Representer Theorems All previous formulations satisfy a multi-task representer theorem ŵ t = n s=1 m i=1 c (t) si x si t {1,..., n} (R.T.) Consequently, a nonlinear kernel can be used All tasks are involved in this expression (unlike the single-task representer theorem Frobenius norm regularization) Generally, consider any matrix optimization problem of the form min w 1,...,w n IR d n t=1 m E ( w t, x ti, y ti ) + Ω(W) i=1 36

38 Representer Theorems (contd.) Definitions: S n + = the positive semidefinite cone The function h : S n + IR is matrix nondecreasing, if h(a) h(b) A, B S n + s.t. A B Theorem. Rep. thm. (R.T.) holds if and only if there exists a matrix nondecreasing function h : S n + IR such that Ω(W) = h(w W) W IR d n (under differentiability assumptions) 37

39 Representer Theorems (contd.) Corollary. The standard rep. thm. for single-task learning (n = 1), ŵ = m c i x i i=1 holds if and only if there exists a nondecreasing function h : IR + IR such that Ω(w) = h( w, w ) w IR d Sufficiency of the condition has been known [Kimeldorf & Wahba, 1970, Schölkopf et al., 2001 etc.] 38

40 Implications Kernelization In single-task learning, the choice of h does not matter essentially However, in multi-task learning, the choice of h is important (since is a partial ordering) Many valid regularizers: Schatten L p norms p, rank, orthogonally invariant norms, norms of type W WM p etc. In matrix learning, kernels and sparsity can be exploited in the same model 39

41 Connection to Learning the Kernel Recall problem (MT L) min D 0, tr(d) 1 min w 1,...,w n IR m n t=1 m E ( w t, x ti, y ti ) + γ w t, D 1 w t i=1 (MT L) learning a common kernel K(x, x ) = x,dx within the convex hull of an infinite number of linear kernels Extends formulation of [Lanckriet et al. 2004] (single task) min K K min c IR m m E ( ) (Kc) i, y i + γ c,kc i=1 in which K was a polytope (convex hull of a finite set of kernels) 40

42 A General Framework for Learning the Kernel Convex set K is generated by basic kernels: K = conv(b) Example 1: Finite set of basic kernels (aka multiple kernel learning ) Example 2: Linear basic kernels B(x, x ) = x, Dx where D bounded, convex set (e.g. (MT L)) Example 3: Gaussian basic kernels B(x, x ) = e x x,σ 1 (x x ) where Σ belongs in a convex subset of the p.s.d. cone 41

43 Learning the Kernel and Structured Sparsity Interpretation of LTK in the feature space [Bach et al. 2004, Micchelli & Pontil 2005] min v 1,...,v N IR m m E i=1 N v j, Φ j (x i ), y i + γ j=1 N v j j=1 2 Group Lasso in the feature space The 2,1 norm tends to favor a small number of feature maps / kernels in the solution 42

44 Why Learn Kernels in Convex Sets? Data fusion (e.g. in bioinformatics) Kernel hyperparameter learning (e.g. Gaussian kernel parameters) Multi-task learning Metric learning Semi-supervised learning (learning the graph) Efficient alternative to cross validation; exploits the power of convex optimization 43

45 Properties of the Solution of LTK Formulation for learning the kernel min K K min c IR m m E ( ) (Kc) i, y i + γ c,kc (LT K) i=1 where K = conv(b) I.e. solutions of (LT K) are of the form ˆK = N ˆλ i 0, N ˆλ i = 1, ˆBi B i=1 i=1 ˆλ i ˆBi, where Any solution (ĉ, ˆK) of (LT K) is a saddle point of a minimax problem 44

46 Properties of the Solution of LTK (contd.) Theorem. (ĉ, ˆK) solves (LT K) if and only if 1. ĉ, ˆB i ĉ = max B B ĉ,bĉ 2. ĉ is the solution to min c IR m i = 1,..., N m E ( ) (ˆKc) i, y i + γ c, ˆKc i=1 Moreover, there exists a solution involving at most m + 1 kernels: ˆK = m+1 i=1 ˆλ i ˆBi 45

47 A General Algorithm for Learning the Kernel Incrementally builds an estimate of the solution, K (k) = k Initialization: Given an initial kernel K (1) in the convex set K while convergence condition is not true do m 1. Compute ĉ = argmin E ( ) (K (k) c) i, y i + γ c,k (k) c c IR m i=1 i=1 λ i K (i) 2. Find a basic kernel ˆB argmax ĉ,bĉ B B 3. Compute K (k+1) as the optimal convex combination of ˆB and K (k) end while 46

48 A General Algorithm for Learning the Kernel (contd.) Theorem. There exists a limit point of the LTK algorithm and any limit point is a solution of (LT K). Step 1 is standard regularization (SVM, ridge regression etc.) Step 2 is the hardest: tractable for e.g. finite B or MTL; non-convex for e.g. Gaussian kernels Step 3 is convex in one variable (requires solving a few regularization problems) Some non-convex cases (e.g. few-parameter Gaussians) are solvable 47

49 Character Recognition Experiments MNIST classification of digits. 1st experiment: Gaussian basic kernels with one parameter σ. Compared with continuously parameterized + local search finite grid of basic kernels SVM 2nd experiment: Gaussian basic kernels with two parameters σ 1, σ 2 (left, right halves of images). Compared with varying finite grids. 48

50 Character Recognition Experiments (contd.) Method Task LTK local finite SVM LTK local finite SVM LTK local finite SVM σ [75, 25000] σ [100, 10000] σ [500, 5000] odd-even vs vs Task LTK LTK LTK σ [75, 25000] σ [100, 10000] σ [500, 5000] odd-even vs vs Using two parameters improves performance Continuous vs. finite grid more efficient robust wrt. parameter range does not overfit 49

51 Character Recognition Experiments (contd.) Learned kernel coefficients. From left to right: odd vs. even (1 and 2 kernel params.), 3 vs. 8 and 4 vs. 7 (2 params.) 50

52 Learning the Graph in Semi-Supervised Learning We can also apply LTK when there are few labeled data points A number of graphs are given, e.g. generated using k-nn, exponential decay weights etc. To exploit the structure of each graph, the Gram matrices B i are taken to be the pseudoinverses of the graph Laplacians 51

53 Conclusion Multi-task learning is ubiquitous; exploiting task relatedness can enhance learning performance significantly Proposed an alternating algorithm to learn tasks that lie on a common subspace; this algorithm is simple and efficient Nonlinear kernels can be introduced via a multi-task representer theorem; also proposed spectral and non-convex matrix learning Multi-task learning can be viewed as an instance of learning combinations of infinite kernels Generally, we can use a greedy incremental algorithm to learn combinations of (finite or infinite) kernels 52

54 Future Work Convergence rates for the algorithms proposed Task relatedness can be of many types (hierarchical features, input transformation invariances etc.) Convex relaxation techniques for the non-convex formulations presented Algorithms and representer theorems for special types of norms Related problems (sparse coding, structured output, metric learning etc.) 53

A Spectral Regularization Framework for Multi-Task Structure Learning

A Spectral Regularization Framework for Multi-Task Structure Learning Massimiliano Pontil Department of Computer Science University College London (Joint work with A. Argyriou, T. Evgeniou, C.A. Micchelli,